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184_notes:resistivity [2018/06/19 14:41] – [Conductivity] curdemma | 184_notes:resistivity [2021/02/27 04:07] (current) – [Making sense of $R$] bartonmo | ||
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Section 19.2 in Matter and Interactions (4th edition) | Section 19.2 in Matter and Interactions (4th edition) | ||
- | [[184_notes: | + | /*[[184_notes: |
- | [[184_notes: | + | [[184_notes: |
===== Resistors and Conductivity ===== | ===== Resistors and Conductivity ===== | ||
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====Resistance==== | ====Resistance==== | ||
- | Before when we talked about resistors, we said that a resistor was a section or part of the circuit where the passage of electrons requires more energy (conventionally, | + | [[184_notes: |
- | {{ 184_notes:resistorshape.jpg?350}} | + | [{{ 184_notes:resistor_shape.png?350|A piece of a resistor with a potential difference of $\Delta$ V from one end to the other, a length L, and a cross-sectional area of A.}}] |
- | == Derivation of $R$ == | + | |
+ | ==== Derivation of $R$ ==== | ||
For example, suppose we have a resistor that has a cross sectional area of $A$, a length $L$, and a potential difference of $\Delta V$ from one end to the other. If we //__assume a steady state current__//, | For example, suppose we have a resistor that has a cross sectional area of $A$, a length $L$, and a potential difference of $\Delta V$ from one end to the other. If we //__assume a steady state current__//, | ||
$$\Delta V =- \int_i^f \vec{E} \cdot \vec{dl}$$ | $$\Delta V =- \int_i^f \vec{E} \cdot \vec{dl}$$ | ||
- | {{184_notes: | + | [{{ 184_notes:resistor_efield_dl.png?300|Electric field direction in a resistor (shown by the red arrow) and the dl vector shown by the blue arrow.}}] |
Because $\vec{E}$ would point along the length of the wire, we would want to integrate along the length of the wire, which would mean that $\vec{E}$ and $\vec{dl}$ would be parallel. This simplifies the dot product to just a multiplication, | Because $\vec{E}$ would point along the length of the wire, we would want to integrate along the length of the wire, which would mean that $\vec{E}$ and $\vec{dl}$ would be parallel. This simplifies the dot product to just a multiplication, | ||
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$$R =\frac{L}{\sigma A}$$ | $$R =\frac{L}{\sigma A}$$ | ||
- | == Making sense of $R$ == | + | ==== Making sense of $R$ ==== |
Why does the bottom fraction make sense? A longer, thinner wire should be more resistive, so the geometric properties make sense (directly proportionally to $L$ and inversely proportional to $A$). A wire with higher conductivity should be less resistive, which also make sense (inversely proprtional to $\sigma$). | Why does the bottom fraction make sense? A longer, thinner wire should be more resistive, so the geometric properties make sense (directly proportionally to $L$ and inversely proportional to $A$). A wire with higher conductivity should be less resistive, which also make sense (inversely proprtional to $\sigma$). | ||
- | Resistance has units of volts per amp, which is also called an ohm. An ohm is represented by a capital omega ($\Omega$). Sometimes you may see resistance rewritten in terms of **resistivity**($\rho$), | + | **Resistance has units of volts per amp, which is also called an ohm.** An ohm is represented by a capital omega ($\Omega$). Sometimes you may see resistance rewritten in terms of **resistivity**($\rho$), |
- | === Ohm's Model === | + | ==== Ohm's Model ==== |
Perhaps equally as important, we can now relate the change in electric potential over a resistor to the resistance and current passing through the resistor. This model of resistance works well for low voltage and currents. This model is also often called " | Perhaps equally as important, we can now relate the change in electric potential over a resistor to the resistance and current passing through the resistor. This model of resistance works well for low voltage and currents. This model is also often called " | ||
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^ Micro ^ Macro ^ | ^ Micro ^ Macro ^ | ||
| $v_{avg}=uE$ | | $v_{avg}=uE$ | ||
- | | $i=nAv_{avg}=nAUE$ | | + | | $i=nAv_{avg}=nAuE$ | |
==== Examples ==== | ==== Examples ==== |